Is the Infinite Derivative of a Function Equal to Zero?

[Dauden]

I was just thinking about this earlier today...

Is the derivative of infinity equal to zero?

Then the more interesting question, is there such thing as an infinite derivative? (I battled with Latex for 20 minutes and can't figure out how to work it so bare with me)

d^inf/dx^inf

And assuming it does exist and that (d^inf/fx^inf) of any x^p is zero where p is a positive integer. Would the infinite derivative of x^inf equal zero?

Has anyone else come across anything like this or am I just assuming too much about the derivative itself?

 

[Yuqing]

Derivatives are rates of change. Taking the derivative of infinity is simply non-sense since infinity is a limit, not a function.

 

[CRGreathouse]

Your language is confused. Are you taking the derivative of infinity (in which case we need a definition of "infinity" as a function), or are you taking the derivative an infinite number of times as suggested by your notation d^inf/fx^inf?

For the latter, I can interpret it thus: define $$\frac{d^\infty}{dx^\infty}f(x)=\lim_{n\to\infty}\frac{d^n}{dx^n}f(x)$$. With that definition, $$\frac{d^\infty}{dx^\infty}x^7=0$$ but $$\frac{d^\infty}{dx^\infty}e^x=e^x$$.

For the former, if you define $$\infty(x)=\infty$$ (on, say, the Riemann sphere) then $$\infty'(x)=0$$.

As to your mention of $$x^\infty$$, I don't know what that means. Formally, I could write $$\frac{d^\infty}{dx^\infty}x^\infty=\infty!$$ but that doesn't have meaning in any structure I can think of.

 

[arildno]

As to your mention of $$x^\infty$$, I don't know what that means. Formally, I could write $$\frac{d^\infty}{dx^\infty}x^\infty=\infty!$$ but that doesn't have meaning in any structure I can think of.

I disagree with your meaningless expression, CRGreathouse.

To me, the proper meaningless expression should be:
$$\frac{d^\infty}{dx^\infty}x^\infty=\infty^{\infty}x^{\infty}$$

 

[Mentallic]

I still find CRGreathouse's meaningless expression more mathematically sound

Dauden, it's like asking what $$\infty -\infty$$ is equal to. It could be zero, it could be finite, or it could be $\pm \infty$. In other words, since we don't know the answer to this, we can't tell you what the answer to the infinite derivative of a polynomial of infinite degree is.

 

[CRGreathouse]

I disagree with your meaningless expression, CRGreathouse.

To me, the proper meaningless expression should be:
$$\frac{d^\infty}{dx^\infty}x^\infty=\infty^{\infty}x^{\infty}$$

That interpretation, like 0, is every bit as meaningful as my result.

 

[arildno]

I still find CRGreathouse's meaningless expression more mathematically sound

Yes. the sweet, sickly odour of a decomposing, dead corpse is still clinging to his expression; mine is utterly desiccated, and hence more pleasant to hang around with...